Probably the Imaskari would have been the first culture to transverse the continent via that route.

But nobody built it, it doesn't have a surface, and it's not maintained because it's not a literal road. There may be roads along the way that make up part of the route, but that's not what the name refers to.

Even if it is not meant to be a literal road, I would add it is still a physical trail, I mean if caravans travel in it every day, at the very minimal you will see treaded ground among the grass or the sand.
However it could be still possible to lose the road if some big natural event cancels the traces, such as a sandstorm or a flood.
Maybe the road could have signposts to indicate direction, if it is heavily travelled. The real silk road had several routes, and many miles with actual literal roads, and you encountered cities, caravanserai, villages, town, inns, taverns and signposts along it, so you could probably lose it only in the less populated regions.
So in a way it is maintained probably, by the people who live along it.
See also here: http://forgottenrealms.wikia.com/wiki/Silk_Road
There are also the Spice Road: http://forgottenrealms.wikia.com/wiki/Spice_Road and the Golden Way: http://forgottenrealms.wikia.com/wiki/Golden_Way

I'd assume there are definitely literal roads making up parts of it, though I'm not so sure it's so precise a route that travelers are following each other's wagon tracks, except in cases where many travelers use the same ford or bridge. It's not a single road or trail, just a general route taken by silk traders from Faerun to Kara-Tur. In some cases there will be roads, trails, bridges, and fords reliable and convenient enough that most travelers will use them, in some cases there won't be.

In some places along the route there may be signposts directing travelers to the next city, but in most cases travelers will rely on major landmarks and the sun and stars.

In general, the Silk Road will be better maintained than the Spice Road, which cuts through less populated regions.

Well some historians doubt there was a Silk road at all, meaning they say there are no proof that people at the time perceived such a connection existed, as the term itself was invented by modern historians in the XX century. Also trade from the west to the east was interrupted at times for decades due to wars or hostile rulers, so basically you can do what you want in a FR campaign, deciding how much you want the road to be safe and recognizable, or even known to the locals..

Yeah, if you wanted to you could say there's a single paved road built by the Imaskari or whoever the whole way, but it could also just be a vaguer "due east until you see a mountain" with no visible trail at all. Probably it's a mix of things.

After several hard days of dawn-to-dusk riding, the Silk Road steepens and begins to climb Repo La, Broken Wall Pass, which leads into the conquered kingdom of Khazari. When the dirt road passes beneath the shadows of the towering Twin Guardian Peaks, Nas Trango Shan and Ahs Trango Shan, Yamun Khahan orders an early camp and sends several patrols ahead to scout the path into the mountains.